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<h3 class="heading"><span class="type">Paragraph</span></h3>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\frac{3}{x-3}=\frac{3}{1+x-4}=\frac{3}{1+t}=3 [1-t+t^2-t^3+\cdots]=3 [1-(x-4)+(x-4)^2-(x-4)^3+\cdots],
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(t=x-4\)</span> and the series is convergent for <span class="process-math">\(|t|&lt;1\)</span> or <span class="process-math">\(|x-4|&lt;1\text{.}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
\frac{1}{x+1}=\frac{1}{5+(x-4)}&amp;=\frac{1}{5}\frac{1}{1+\frac{x-4}{5}}=\frac{1}{5}[1-t+t^2-t^3+\cdots]\\
&amp;=\frac{1}{5} \left[ 1-\frac{x-4}{5}+\left(\frac{x-4}{5} \right)^2 -\left(\frac{x-4}{5}\right)^3 +\cdots  \right],
\end{aligned}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(t=(x-4)/5\)</span> and the series is convergent for <span class="process-math">\(\left| \frac{x-4}{5} \right|&lt;1\)</span> or <span class="process-math">\(|x-4|&lt;5\text{.}\)</span> Therefore, <span class="process-math">\(\frac{Q(x)}{P(x)}\)</span> is convergent for <span class="process-math">\(|x-4|&lt;1\)</span> and the radius of convergence is <span class="process-math">\(1\text{.}\)</span> Similarly,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\frac{R(x)}{P(x)}=\frac{4}{(x-3)(x+1)}=\frac{1}{x-3}-\frac{1}{x+1},
\end{equation*}
</div>
<p class="continuation">which is convergent for <span class="process-math">\(|x-4|&lt;1\)</span> and the radius of convergence is <span class="process-math">\(1\text{.}\)</span> According to the theorem, a lower bound for the radius of convergence for the series solution is <span class="process-math">\(1\text{.}\)</span></p>
<span class="incontext"><a href="sec5_3.html#p-216" class="internal">in-context</a></span>
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